Tangents to polar curves

Polar coordinates represent location in terms of radius (r) and angle (θ), rather than x and y used in Cartesian coordinates.
A polar curve is described by an equation in r and θ. It represents points in a plane, the location of which are determined by r and θ.
Tangent lines to polar curves can be found using a specific set of rules, just like for Cartesian curves.
The slope of a tangent to a curve at any given point measures its steepness. In polar coordinates, this means calculating dr/dθ, or how much r changes for a small change in θ.
To find the equation of a tangent line to a polar curve, first differentiate the equation r(θ) with respect to θ to get dr/dθ.
Next, use the formula for the slope of a tangent line in polar coordinates: dy/dx = (rsinθ + cosθdr/dθ)/(rcosθ - sinθdr/dθ).
Note: dy/dx is the slope of the tangent in Cartesian coordinates. You use this because even though you’re dealing with a polar curve, you’re interested in the gradient in the Cartesian x-y plane.
Remember the equation of the tangent line in polar form is: (r_0/r) = cos(θ - θ_0), where r_0 and θ_0 are the polar coordinates of the point where the tangent meets the curve.
Becoming skilled at converting between Cartesian and polar coordinates will be very helpful when dealing with tangents to polar curves.
Also remember that the radian measure is generally the standard unit when dealing with polar coordinates.
The sign of dr/dθ is crucial to correctly interpret the direction in which r changes as θ increases: a positive sign means r is increasing, while a negative sign means r is decreasing.
Also remember the rules for differentiating sine and cosine functions. You will use them often when differentiating polar equations to find dr/dθ.
Practice is the key to mastering this topic. Work through example problems and check your understanding by doing exercises with different rates of change and different polar equations.